A390553 - OEIS

%I #33 Feb 18 2026 02:24:37

%S 1,1,4,142,163541,11402245766,73126631811200332,

%T 61813796692992154901350387,9377589491378940028767495513551218152,

%U 333878364642348231785641974791552269083712281908056,3534887605489498667394663017655202046325566383504435047753396139331

%N Total number of permutations of [k] having exactly n cycles with cycle length set [m] for nonnegative integers k and m.

%C The values of k and m can be restricted: n <= k <= n*(n+1)/2 and 0 <= m <= n. Other values do not contribute positively. Note that [0] = {}, [1] = {1}, [k] = {1..k}.

%H Alois P. Heinz, <a href="/A390553/b390553.txt">Table of n, a(n) for n = 0..30</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>

%F a(n) = Sum_{k=n..n*(n+1)/2} A392215(k,n).

%e a(2) = 4: (1)(2), (1)(23), (13)(2), (12)(3).

%p b:= proc(n, i) option remember; `if`(n=0, 1, add(expand(b(n-i*j, i+1))*

%p       x^j*combinat[multinomial](n, n-i*j, i$j)/j!*(i-1)!^j, j=1..n/i))

%p     end:

%p a:= n-> add(coeff(b(k, 1), x, n), k=n..n*(n+1)/2):

%p seq(a(n), n=0..10);

%t b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[Expand[b[n-i*j, i+1]]*x^j*Multinomial@@Join[{n-i*j}, Table[i, {j}]]/j!*(i-1)!^j, {j, 1, n/i}]];

%t a[n_] := Sum[Coefficient[b[k, 1], x, n], {k, n, n(n+1)/2}];

%t Table[a[n], {n, 0, 10}] (* _Jean-François Alcover_, Feb 17 2026, after _Alois P. Heinz_ *)

%Y Column sums of A392215.

%Y Cf. A000110, A000217.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Feb 03 2026